The inverse spectral problem for first order systems on the half line

On the half line [0,∞) we study first order differential operators of the form B 1 i d dx +Q(x), where B := ( B1 0 0 −B2 ) , B1, B2 ∈ M(n,C) are self–adjoint positive definite matrices and Q : R+ → M(2n,C), R+ := [0,∞), is a continuous self–adjoint off–diagonal matrix function. We determine the self–adjoint boundary conditions for these operators. We prove that for each such boundary value problem there exists a unique matrix spectral function σ and a generalized Fourier transform which diagonalizes the corresponding operator in L2σ(R,C). We give necessary and sufficient conditions for a matrix function σ to be the spectral measure of a matrix potential Q. Moreover we present a procedure based on a Gelfand-Levitan type equation for the determination of Q from σ. Our results generalize earlier results of M. Gasymov and B. Levitan. We apply our results to show the existence of 2n×2n Dirac systems with purely absolute continuous, purely singular continuous and purely discrete spectrum of multiplicity one.